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Theorem nmoofval 28076
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSet‘𝑈)
nmoofval.2 𝑌 = (BaseSet‘𝑊)
nmoofval.3 𝐿 = (normCV𝑈)
nmoofval.4 𝑀 = (normCV𝑊)
nmoofval.6 𝑁 = (𝑈 normOpOLD 𝑊)
Assertion
Ref Expression
nmoofval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀
Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧,𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (𝑈 normOpOLD 𝑊)
2 fveq2 6377 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2817 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 6860 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6377 . . . . . . . . . . 11 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCV𝑈)
86, 7syl6eqr 2817 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = 𝐿)
98fveq1d 6379 . . . . . . . . 9 (𝑢 = 𝑈 → ((normCV𝑢)‘𝑧) = (𝐿𝑧))
109breq1d 4821 . . . . . . . 8 (𝑢 = 𝑈 → (((normCV𝑢)‘𝑧) ≤ 1 ↔ (𝐿𝑧) ≤ 1))
1110anbi1d 623 . . . . . . 7 (𝑢 = 𝑈 → ((((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
124, 11rexeqbidv 3301 . . . . . 6 (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
1312abbidv 2884 . . . . 5 (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))})
1413supeq1d 8561 . . . 4 (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
155, 14mpteq12dv 4894 . . 3 (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
16 fveq2 6377 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 nmoofval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17syl6eqr 2817 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918oveq1d 6859 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
20 fveq2 6377 . . . . . . . . . . 11 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCV𝑊)
2220, 21syl6eqr 2817 . . . . . . . . . 10 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
2322fveq1d 6379 . . . . . . . . 9 (𝑤 = 𝑊 → ((normCV𝑤)‘(𝑡𝑧)) = (𝑀‘(𝑡𝑧)))
2423eqeq2d 2775 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((normCV𝑤)‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑡𝑧))))
2524anbi2d 622 . . . . . . 7 (𝑤 = 𝑊 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2625rexbidv 3199 . . . . . 6 (𝑤 = 𝑊 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2726abbidv 2884 . . . . 5 (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))})
2827supeq1d 8561 . . . 4 (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))
2919, 28mpteq12dv 4894 . . 3 (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
30 df-nmoo 28059 . . 3 normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
31 ovex 6876 . . . 4 (𝑌𝑚 𝑋) ∈ V
3231mptex 6681 . . 3 (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpt2 6996 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
341, 33syl5eq 2811 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  wrex 3056   class class class wbr 4811  cmpt 4890  cfv 6070  (class class class)co 6844  𝑚 cmap 8062  supcsup 8555  1c1 10192  *cxr 10329   < clt 10330  cle 10331  NrmCVeccnv 27898  BaseSetcba 27900  normCVcnmcv 27904   normOpOLD cnmoo 28055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-sup 8557  df-nmoo 28059
This theorem is referenced by:  nmooval  28077  hhnmoi  29219
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